The Magnetic Genome of Two-Dimensional van der Waals Materials

Abstract

Magnetism in two-dimensional (2D) van der Waals (vdW) materials has recently emerged as one of the most promising areas in condensed matter research, with many exciting emerging properties and significant potential for applications ranging from topological magnonics to low-power spintronics, quantum computing, and optical communications. In the brief time after their discovery, 2D magnets have blossomed into a rich area for investigation, where fundamental concepts in magnetism are challenged by the behavior of spins that can develop at the single layer limit. However, much effort is still needed in multiple fronts before 2D magnets can be routinely used for practical implementations. In this comprehensive review, prominent authors with expertise in complementary fields of 2D magnetism (i.e., synthesis, device engineering, magneto-optics, imaging, transport, mechanics, spin excitations, and theory and simulations) have joined together to provide a genome of current knowledge and a guideline for future developments in 2D magnetic materials research.

Keywords: 2D magnetic materials; CrI3; Fe3GeTe2; atomistic spin dynamics; magnetic genome; magneto-optical effect; neutron scattering; van der Waals.

Figure 1

Timeline of developments in 2D magnets. Since early 2016, a few results on monolayer phosphides MPX₃ (M = Fe, Mn, Ni, Cd; X = S, Se)^, and CrSiTe₃ appeared in the literature, with results on electron tunneling in MnPS₃ also being reported. The conclusive measurements in 2017 of magnetism on CrI₃ and Cr₂Ge₂Te₆ sparked an increasing interest in several subjects involving magnetism in 2D. Results on spin–lattice coupling collected from CrCl₃ also provided different mechanisms involving vibrations and spins in 2D. In 2018, the electric control of magnetism,⁻ giant magnetoresistance,⁻ and a potential 2D magnet (i.e., VSe₂) displaying room-temperature magnetism⁻ attracted substantial interest in the community. In 2019, experimental evidence of stacking-dependent magnetic properties,^, pressure effects,^, and giant second-harmonic generation (SHG) drove the field toward intriguing magnetic properties. In 2020, spin-textures⁻ such as skyrmions, spirals, and spin-waves indicate that topologically nontrivial spins are a reality on 2D magnets. In 2021, a few reports on twisted magnetic layers,^, together with the hybrid character of narrow domain-walls on CrI₃, raised possibilities for the angular control of magnetic features and domain-wall based applications (i.e., racetrack). All images adapted from the references cited above with permission as follows. Panels from (2016) reprinted with permission from ref (32), copyright 2016 American Chemical Society; ref (3), copyright 2016 Royal Society of Chemistry; ref (1), copyright 2016 American Chemical Society; and ref (4), copyright 2016 AIP Publishing and reprinted with permission under a Creative Commons Attribution (CC BY) license. Panels from (2017) reprinted with permission from ref (5), copyright 2017 Springer Nature; ref (6), copyright 2017 Springer Nature; and ref (7), copyright 2017 American Physical Society. Panels from (2018) reprinted with permission from ref (8), copyright 2018 Springer Nature; ref (9), copyright 2018 Springer Nature; ref (13), copyright 2018 AAAS; with permission under a Creative Commons CC by 4.0 license from ref (15), copyright 2018 Springer Nature; and ref (17), copyright 2018 American Chemical Society. Panels from (2019) reprinted with permission from ref (22), copyright 2019 Springer Nature; ref (23), copyright 2019 Springer Nature; ref (20), copyright 2019 AAAS; ref (21), copyright 2019 AAAS; and ref (24), copyright 2019 Springer Nature. Panels from (2020) reprinted with permission from ref (25), copyright 2020 American Chemical Society; and ref (28), copyright 2020 Springer Nature. Panels from (2021) reprinted with permission from ref (29), copyright 2021 Springer Nature; ref (30), copyright 2021 Springer Nature; and ref (31), copyright 2021 John Wiley and Sons.

Figure 2

Role of spin dimensionality n on the 3D-axis. (a) n = 1; 1D Ising type, where spins point in either up or down along a given direction (e.g., easy axis). (b) n = 2; 2D XY type, where spins are constrained to a given plane (easy-plane anisotropy) without any restriction on which plane (e.g., XY, XZ, YZ). (c) n = 3; 3D isotropic Heisenberg type, where spins have no constraints on the direction assuming any position along the 3D sphere.

Figure 3

Routes toward room-temperature FM ordering in 2D layered materials by (a) electrostatic gating, (b) tuning dimensionality, and (c) proximity effect. Representative experimental examples include: (a) ionic liquid gating in Cr₂Ge₂Te₆, achieving a dramatic increase from 60K to 180 K in the ordering temperature. Adapted with permission from ref (86). Copyright 2020 Springer Nature. (b) Substantial T_c enhancement in Cr₂Te₃ flakes as the thickness is reduced, measured by anomalous Hall effect. Adapted with permission from ref (80). Copyright 2020 American Chemical Society. (c) Persistent magnetic signals up to 380 K in 4 nm-thick Fe₃GeTe₂ interfaced with the topological insulator Bi₂Te₃. Adapted with permission from ref (90). Copyright 2020 American Chemical Society.

Figure 4

Theoretical calculations for the Curie temperature in TMD Janus systems. (a) Diagram of the VSeTe crystal structure in top-down and cross-sectional views. (b) Temperature-dependent magnetic moment and specific heat of VSeTe, obtained via Monte Carlo simulations in a nearest-neighbor Heisenberg exchange model. Panels (a) and (b) reprinted with permission from ref (97). Copyright 2009 Royal Society of Chemistry. (c) Curie temperatures of V-, Cr- and Mn-based Janus TMDs, highlighting VSSe and VSeTe as the candidates with highest T_c. Adapted with permission from ref (96). Copyright 2018 Elsevier.

Figure 5

Examples of large-area 2D magnets grown by MBE. (a–d) Monolayer CrCl₃ on Graphene/SiC(0001) and (e–g) few-layer Fe₃GeTe₂ on GaAs (111). (a) Schematic crystal structure of CrCl₃/graphene/6H-SiC layers in top-down view and cross-section view. (b) In situ RHEED pattern of the substrate and monolayer CrCl₃ grown by MBE, along Γ–M of graphene (Γ–K of SiC). Streaks from different high-symmetry directions of CrCl₃ are observed, implying a twisted in-plane orientation of the grains. (c) STM topography of a monolayer CrCl₃ grown on graphene/6H-SiC(0001), indicating a homogeneous coverage on long length scales. Inset: A magnified topography image, which reveals the grain boundaries. (d) Atom resolved image of the CrCl₃ lattice featuring a moiré pattern (upper panel) and its Fourier transformed image (lower panel). Panels (a–d) adapted with permission from ref (78). Copyright 2021 AAAS. (e) Crystal structure of Fe₃GeTe₂. (f) RHEED oscillations indicating layer-by-layer growth of Fe₃GeTe₂ (0001) on GaAs (111), and the corresponding electron diffraction pattern (inset). The inferred growth rate is 111 s per monolayer. (g) Transmission electron microscopy of a Fe₃GeTe₂/GaAs cross-section, indicating the (111)/(0002) epitaxial relationship. Panels (e–g) adapted with permission under a Creative Commins CC BY license from ref (99). Copyright 2017 Springer Nature.

Figure 6

Enormous TMR in vdW MTJs induced by layered antiferromagnetism. (A) Reflective magnetic circular dichroism (RMCD) of bilayer (left) and trilayer (right) CrI₃ showing layer-dependent magnetism. Reproduced with permission from ref (13). Copyright 2018 AAAS. (B) Magnetic-field-dependent current change in a vdW MTJ incorporating magnetic CrI₃. (C) Schematic energy diagrams of CrI₃-based MTJs with AF barrier (top) and FM barrier (bottom). Panels (B and C) are adapted with permission from ref (16). Copyright 2018 American Chemical Society. (D) Summary of TMR values as a function of thickness. Reproduced from ref (112). Copyright 2019 American Chemical Society.

Figure 7

Summary of intrinsic magnetoresistance in vdW magnets. (A) Magnetoresistance ratio MRR(B) in bulk CrSBr versus magnetic field (parallel to the b-axis) at various temperatures. Each MRR curve is offset for clarity. The solid black line is an MRR curve taken near the Néel temperature. The antiferromagnetic (AF), fully polarized (FP), and paramagnetic (PM) phases are labeled, and the phase boundary is denoted by dashed black lines. Schematics showing the orientation of the spins in the AF and FP states are given above the plot. Reproduced with permission from ref (110). Copyright 2020 John Wiley and Sons. (B) R_yx of a 5-layer MnBi₂Te₄ sample as a function of external magnetic field applied perpendicular to the sample plane at T = 1.6 K. Data are symmetrized to remove the R_xx component. (C) R_xx of a 5-layer MnBi₂Te₄ flake as a function of magnetic field acquired at various temperatures. Data are symmetrized to remove the R_yx component. Inset shows the layered crystal structure of MnBi₂Te₄ in the AF state. Panels (B) and (C) are reproduced with permission from ref (111). Copyright 2020 AAAS. (D) Ball and stick model of the Cr₂Ge₂Te₆ crystal structure. (E) Magnetoresistance curves for T = 60 K and back-gate voltage of 3.9 V for a 22 nm-thick Cr₂Ge₂Te₆ flake. The background is removed for clarity. The magnetic field is applied in the out-of-plane direction. Unprocessed data are shown in the inset. Panels (D) and (E) are reproduced with permission from ref (86). Copyright 2020 Springer Nature. (F) Side view of the atomic lattice of bilayer Fe₃GeTe₂. The dashed rectangular box denotes the crystal unit cell. (G) Temperature-dependent magnetic field (out-of-plane) sweeps of the Hall resistance measured on a 12 nm thick Fe₃GeTe₂ flake. Panels (F) and (G) are reproduced with permission from ref (77). Copyright 2018 Springer Nature.

Figure 8

Comparison between tunnelling differential conductance spectra between Gr/CrI₃/Gr and Gr/CrBr₃/Gr devices in a magnetic field applied out-of-plane of the magnetic film for CrI₃ sample and in-the-plane of the magnetic film for CrBr₃ sample. For both materials, narrow low energy resonances appear, which are dispersive in the magnetic field. The color maps demonstrating the dependence of the differential conductance for (a, b) Gr/CrI₃/Gr and (c) Gr/CrBr₃/Gr devices illustrate the evolution of such resonances. The quantitative parameters describing the magnonic states, i.e., their zero-field energy and the slope ΔE/ΔB can be extracted based on the linear fits to the evolution of the energy of the resonances in the magnetic field (b, d). Panels (a) and (b) are adapted with permission from ref (14). Copyright 2018 AAAS. Panels (c) and (d) are adapted with permission from ref (131). Copyright 2018 Springer Nature.

Figure 9

Temperature evolution of the differential conductance spectra in an out-of-plane magnetic field of 17.5 T is indicative of elastic and inelastic tunnelling processes mediated by magnons. The experimental data (a) shows qualitative agreement with theoretical predictions (b), which consider inelastic scattering processes with an emission of a magnon and elastic tunnelling processes involving two magnons. Both processes can be distinguished by applying a bias which shifts the mutual alignment of Landau levels (LLs) in the graphene electrodes (c, d). All panels adapted with permission from ref (131). Copyright 2018 Springer Nature.

Figure 10

(a) Schematic illustration in a single-particle picture of the direct band gap edge states for a W-based TMD (bottom) in the degenerate but inequivalent corners of the hexagonal Brillouin zone (top). The red dashed (blue solid) lines depict spin-up (down) band-edge states. Up (down) short arrows indicate spin-up (down) conduction band and valence band electrons. Long vertical arrows represent spin-allowed optical transitions with σ⁺ (red) and σ^– (blue) polarization. Adapted with permission from ref (146). Copyright 2018 American Physical Society. (b) Schematic illustration of the optical response of an ideal 2D semiconductor, showing the exciton ground (n = 1) and excited state resonances (n = 2, 3, 4, ...) below the renormalized quasiparticle band gap. Adapted with permission under a Creative Commons CC BY 4.0 license from ref (147). Copyright 2017 Springer Nature. The top right inset shows the energy level scheme of the exciton states, designated by their principal quantum number n, with the binding energy of the exciton ground state below the free-particle (FP) band gap. Adapted with permission from ref (146). Copyright 2018 American Physical Society. (c) Schematic energy level diagram depicting the three contributions to the valley Zeeman shifts of the band-edge states: spin (black), valley (green) and atomic orbital (purple). The dashed (solid) lines represent the energies of the states before (after) applying a positive magnetic field perpendicular to the material interface. Adapted with permission from ref (148). Copyright 2015 Springer Nature. (d) Schematics of a typical microscope for optical spectroscopy of 2D materials in epifluorescence geometry. The 2D materials can be studied at temperatures T = 4–300 K by placing them on nonmagnetic nanopositioners inside a cryostat. A solenoid allows the application of magnetic fields (B) perpendicular to the crystal plane (Faraday geometry). The excitation and collection paths can feature several polarizing optical components for PL and reflectance experiments in circular (σ) and linear (π) bases. Adapted with permission from ref (149). Copyright 2017 Springer Nature. (e) PL spectrum of ML MoSe₂ at T = 4 K under continuous-wave laser excitation with 2.33 eV. The spectrum shows the neutral exciton (X⁰) and the lower-energy negatively charged exciton (X^–). The X^– shows a binding energy of about 30 meV (see inset). Adapted with permission from ref (150). Copyright 2013 Springer Nature. (f) Derivative of the reflectance contrast spectrum (d/dE) (ΔR/R) for a WS₂ monolayer (ML). The exciton states are labeled by their respective quantum numbers. The inset shows the as-measured reflectance contrast ΔR/R for comparison. Adapted with permission from ref (151). Copyright 2014 American Physical Society.

Figure 11

(a) Schematic band structure depicting intravalley (left) and intervalley (right) interactions between an exciton and a spin-polarized 2DES in a lightly doped W-based ML TMD. Adapted with permission from ref (170). Copyright 2020 AIP Publishing. (b) Voltage-gate-dependent color-scale map presenting reflectance contrast spectra measured in an hBN-encapsulated ML MoSe₂. Attractive and repulsive exciton polarons are visible in both n- and p-doped regimes. Adapted with permission under a Creative Commons CC BY 4.0 license from ref (171). Copyright 2019 American Physical Society. (c) Gate-voltage dependencies of the line widths (top panel) and energies (middle panel) of the exciton (red) and attractive polaron (blue) resonances on the hole-doped side of an hBN-encapusulated ML MoSe₂ at B = 16 T. Bottom panel: Gate voltages corresponding to integer filling factors determined based on the positions of the local minima of the exciton line width. Adapted with permission under a Creative Commons CC BY 4.0 license from ref (171). Copyright 2019 American Physical Society. (d) Gate-voltage-dependence derivative of the reflectance contrast spectrum with respect to the gate voltage of a charge-tunable, dual-graphene-gated, and fully hBN-encapsulated MoSe₂ ML (right panel). The weak, higher-energy resonance is due to Umklapp scattering of the excitons off the electron Wigner crystal. Left panel: Schematics of the exciton dispersion in a ML TMD semiconductor hosting an electron system in various structural phases. The parabolic- and linear-in-momentum exciton branches arise from the splitting of the exciton branches due to the electron–hole exchange interaction, and correspond to the exciton dipole oriented along transverse (T) or longitudinal (L) directions with respect to the momentum vector, respectively. Adapted with permission from ref (172). Copyright 2021 Springer Nature. (e) Top panels: schematics of a Wigner crystal in a MoSe₂ bilayer with an intercalated 1 nm thick layer of hBN (left). The top right panels show schematics of commensurate stacking in bilayer Wigner crystals with triangular lattices for filling ratios n_t:n_b of 1:1, 4:1, and 7:1, with n_t and n_b the carrier density in the top (blue dots) and bottom (red dots) MoSe₂ layers. Bottom panels: 2D map of δ(n_t, n_b) as a function of total carrier density n and temperature T for n_t:n_b = 1:1 (right). The Wigner crystal forms in the region δ > 0 region. Theoretical schematic phase diagram of a bilayer Wigner crystal, showing both quantum and thermal phase transitions (left). Adapted with permission from ref (173). Copyright 2021 Springer Nature.

Figure 12

(a) Schematic illustration of a 2D triangular moiré superlattice resulting from stacking two TMD MLs with different lattice constants and/or twist angle. The moiré potentials (empty blue circles) can be loaded with either electrons or holes with spin up (red arows) or down (blue arrows). The on-site Coulomb interaction energy U and hopping amplitude t between spins in the lattice is highly tunable by the stacking angle and choice of 2D materials, enabling the investigation of the Fermi–Hubbard model. (b) Sketch of the type-II band structure of a WSe₂/WS₂ heterobilayer, where K and K′ represent two valley degrees of freedom. Up (down) arrows denote the spin up (down) direction. (c) Dependence of the magnetic susceptibility χ ∝ g – g₀ (left axis, black filled symbols) and Weiss constant θ (right axis, red empty symbols) on the filling factor of WSe₂/WS₂ at 1.65 K. Panels (b) and (c) adapted with permission from ref (197). Copyright 2020 Springer Nature. (d) Optically detected resistance and capacitance signal at 1 kHz (gray) and 30 kHz (black) from charge-neutral to moderate hole doping in WSe₂/WS₂, showing gap-like features at hole doping levels of n = n₀/3 (orange dashed line), n = 2n₀/3 (green dashed line) and n = n₀ (blue dashed line). Adapted with permission from ref (198). Copyright 2020 Springer Nature. (e) Abundance of insulating states in WSe₂/WS₂ as revealed by the blue-shifts of the 2s exciton resonance in the reflectance contrast of a ML WSe₂ sensor placed in close proximity to the heterobilayer (top panel). The top axis shows the proposed filling factor for the insulating states, with the corresponding configurations schematically shown in the bottom panels. Adapted with permission from ref (199). Copyright 2020 Springer Nature.

Figure 13

(a) Top: Schematic illustration of electron–hole pairs forming 1s and 2s excitonic states in a 2D dielectric slab. Adapted with permission from ref (151). Copyright 2014 American Physical Society. Bottom: Theoretically calculated energies of the bandgap and exciton states in a WS₂ ML as a function of inverse squared external dielectric constant. Shaded areas indicate fluctuations from variations of the external screening. Adapted with permission from ref (202). Copyright 2019 Springer Nature. (b) Schematic of a device heterostructure to demonstrate the sensing capabilities of excitons in ML WSe₂. (c) Reflectance contrast of the 1s (left) and 2s and 3s (middle) excitonic transitions in the WSe₂ sensor, and the 1s exciton in the WS₂ sample as a function of the applied gate voltage (i.e., the electron concentration in WS₂). (b) and (c) Adapted with permission from ref (199). Copyright 2020 Springer Nature. (d) Left: Reflective magneto-circular dichroism as a function of magnetic field (bottom) measured in a monolayer WSe₂ and trilayer CrI₃ heterostructure, depicted above schematically. Orange and green curves represent magnetic field sweeping up (increase) and down (decrease), respectively. Right: Schematic of a ML WSe₂ and bilayer CrI₃ heterostructure (top). The layered AF spatial domains that are indistinguishable by reflective magneto-circular dichroism can be resolved by circular polarization-resolved PL from WSe₂ (bottom). Adapted with permission from ref (203). Copyright 2020 Springer Nature. (e) Measurement of the CrBr₃ magnetization hysteresis using the MOKE (bottom) in a device like schematically shown on top. Adapted with permission under a Creative Commons CC BY 4.0 license from ref (204). Copyright 2020 American Physical Society.

Figure 14

(a) Top view of CrI₃ monolayer with gray and purple spheres representing Cr and I atoms. Adapted with permission from ref (5). Copyright 2017 Springer Nature. (b, c) Lateral view of bulk CrI₃ in the monoclinic (b) and rhombohedral (c) phase. (d, e) Polarization resolved Raman spectra for bulk CrI₃ taken at 5 K (d) and 280 K (e). (f, g) Polarization resolved Raman spectra for thin layer CrI₃ taken at 5 K (f) and 280 K (g). Panels (b–g) adapted with permission under a Creative Commons CC BY license from ref (224). Copyright 2019 IOP Publishing.

Figure 15

(a) Raman spectra taken on bulk CrI₃ in LL and LR channels at 10 K and 0 T. P(A_g), P(E_g), and M_1,2 label phonon modes of A_g, E_g symmetry and coupled with layered magnetism, respectively. (b) Color map of magnetic field dependent Raman spectra taken in the LL channel. (c) Raman spectra taken on bulk CrI₃ in LL and LR channels at 7 T. (d–f) Magnetic field dependence of selected A_g and E_g phonon modes. (g) A schematic illustration of the monoclinic distortion across the layered surface AF to FM transition. Adapted with permission under a Creative Commons CC BY 4.0 license from ref (218). Copyright 2020 American Physical Society.

Figure 16

(a) Raman spectra taken in both linear parallel and crossed channels in monolayer CrI₃ at 15 K and 0 T. Polar plots of polarization dependent Ag mode intensity (∼129 cm^–1) at 60 and 15 K, above and below the magnetic onset temperature T_C = 45 K, respectively. (b) Raman spectra of 2L CrI₃ at 15 K and at 0 and 1.5 T in linearly parallel and crossed channels. Panels (a, b) adapted with permission from ref (221). Copyright 2020 Springer Nature. (c) Raman spectra of 2L CrI₃ at 1.7 K and at 0 T and −1 T in linearly parallel and crossed channel. Adapted with permissions from ref (222). copyright 2020 American Chemical Society. (d) Raman spectra of 10L CrI₃ at 9 K and at selected magnetic fields. Adapted with permission under a Creative Commons CC BY license from ref (223). Copyright 2020 Springer Nature.

Figure 17

(a) Raman spectra taken on 1–4L CrI₃ in both linear parallel and crossed channels at 10 K and 0 T. Adapted with permission under a Creative Commons CC-BY license from ref (219). Copyright 2017 National Academy of Sciences. (b) A summary of linear chain model calculation results and selection rule analysis for 1–4L CrI₃.

Figure 18

(a) Raman spectra taken on 4L CrI₃ in the RR channel at 10 K and at 0, 1, and 2 T, corresponding to the three magnetic field ranges B < B_c1, B_c1 < B < B_c2, and B > B_c2, respectively. (b) Magnetic field dependence of three modes U_1,3,4. (c) A summary of the analysis of the layered magnetism-assisted Raman scattering for four phonon modes and three layered magnetic structures. (d) List of phonon modes in 4L CrI₃ that contribute to the layered magnetism-assisted scattering channel with antisymmetric Raman tensors. (e) Calculated magnetic field dependence of the four phonons of 4L CrI₃, U_1–4, in the RR channel. Adapted with permission under a Creative Commons CC-BY license from ref (219). Copyright 2017 National Academy of Sciences.

Figure 19

(a) Raman spectra of 2L CrI₃ taken at 10 and 70 K with an excitation wavelength of 633 nm. Solid orange lines are fits to the raw Raman spectra, using a sum of N Lorentzian profiles and a constant background, . (b) Histogram plot of the fitted Lorentzian mode intensity (A_N) as a function of N at 10 and 70 K. Solid curves are fits of the peak intensity profiles to the Poisson distribution functions, . (c) Plot of 2D e-ph coupling constant (α_2D) as a function of temperature. The dashed vertical line marks the magnetic onset T_C = 45 K. Adapted with permission under a Creative Commons CC BY license from ref (220). Copyright 2020 Springer Nature.

Figure 20

(a) Raman spectra of 2L CrI₃ acquired at 10 K in the RR and LL polarization channels with an applied out-of-plane magnetic field (B_⊥) of 1 T (top), 0 T (middle), and −1 T (bottom), respectively. (b, c) Plots of the Poisson fit amplitude A₀ (b) and the 2D e-ph coupling strength α_2D (c) as a function of the applied B_⊥ in the RR (orange data points) and LL (blue) channels. Solid lines are step (orange and blue in b) and linear (gray in c) function fits to the magnetic field dependence of A₀ and α_2D, respectively. Adapted with permission under a Creative Commons CC BY license from ref (220). Copyright 2020 Springer Nature.

Figure 21

Birefringence effects in atomically thin CrI₃. (a) The measurement of Kerr rotation angle in external magnetic field reveals finite magnetization in CrI₃ monolayers. The inspection of CrI₃ multilayers demonstrates AF interlayer coupling in (b) bilayers and (c) trilayers. Panels (a–c) adapted with permission from ref (5). Copyright 2017 by Springer Nature. (d, e) Qualitatively, the same behavior for atomically thin layers is observed when inspecting MCD. Adapted with permission from ref (13). Copyright 2018 AAAS.

Figure 22

Photoluminescence bands in CrI₃ monolayers below Curie temperature display large degree of circular polarization which can be linked to the emergence of magnetization. A rapid reorientation of magnetic moments leads to the change in the sign of the polarization degree combined with a hysteric behavior with a coercive field of about 0.1 T, in agreement with analogous observation by birefringence effects. Adapted with permission from ref (215). Copyright 2017 Springer Nature.

Figure 23

(a) MOKE maps of a CrI₃ monolayer at external magnetic fields of 0, 0.15, and 0.3 T. Adapted with permission from ref (5). Copyright 2017 Springer Nature. (b) Kerr rotation signal for Cr₂Ge₂Te₆ bilayer flake under 0.075 T as the temperature decreases from 40 to 4.7 K. The average background signal has been subtracted and the signals are truncated at 30 μrad. Adapted with permission from ref (6). Copyright 2017 Springer Nature. (c) Pump-induced Kerr rotation as a function of pump–probe delay time in bilayer CrI₃ under different in-plane magnetic fields. Adapted with permission from ref (28). Copyright 2020 Springer Nature. (d) (iii) Illustration of the wide-field MCD experimental setup. Blue and red beams represent illumination light from the laser and scattered light from the sample with different effective numeric apertures. HWP: Half-wave plate. QWP: Quarter-wave plate. (ii–iv) Optical microscopy image (ii) and polarization-enhanced MCD image (iii, iv) of a monolayer CrBr₃ (white dashed box). The MCD image shows giant optical contrast of ±60% for the positive (iii) and negative (iv) remnant magnetization. Adapted with permission from ref (246). Copyright 2020 Springer Nature.

Figure 24

(a) STXM images show the initial labyrinth domain states stabilized at 120 and 100 K transformed to the magnetic skyrmion lattices induced by bipolar pulse bursts. The other two images at B_z = 0 mT were acquired after removing the external fields. (b) Lorentz TEM images of magnetic skyrmion lattices taken at the sample tilting angle of −20° (left), 0° (middle) and 20° (right) with respect to x-axis as illustrated in the upper panels at B_z = −40 mT and 160 K, respectively. Panels (a) and (b) are adapted with permission from ref (26). Copyright 2021 American Physical Society. (c) (i) Overfocused Lorentz-TEM images of the skyrmion bubbles taken at 93 K and in zero-field. (ii) An enlarged in-plane magnetization distribution map obtained by transport of intensity equation analysis for a selected skyrmion bubble indicated by the white dotted box in (i). (iii) Simulation of skyrmion lattices at an out-of-plane field of 60 mT. Adapted with permission from ref (25). Copyright 2020 American Chemical Society. (d) Schematic diagram of a Néel-type skyrmion on a tilt sample for Lorentz TEM imaging (left). The orange and blue circles are for positive and negative magnetizations along z direction, respectively. Brown arrows indicate the in-plane magnetization component, while gray arrows indicate the Lorentz force. Lorentz TEM images (right) observation of skyrmion lattice from under focus to over focus on WTe₂/40L Fe₃GeTe₂ samples at 180 K with a field of 51 mT. Adapted with permission under a Creative Commons CC BY license from ref (27). Copyright 2020 Springer Nature.

Figure 25

(a) STM images of a CrBr₃ film with adjacent monolayer (1L) and bilayer (2L) regions and spin-polarized tunneling signals on the bilayer regions as a function of magnetic field. Adapted with permission from ref (20). Copyright 2019 AAAS. (b) Antiferromagnetic domain patterns in MnBi₂Te₄ measured with MFM. Adapted with permission from ref (253). Copyright 2020 American Chemical Society. (c) Schematic illustration of the pulled quartz tube with two Nb or Pb superconducting leads connected to Au electrodes and image of a Nb SQUID-on-tip device. Inset: Magnified view showing the superconducting loop on the apex of the tip. The bridges that reside in the gap regions between the leads form the two weak links of the SQUID. Adapted with permission from ref (254). Copyright 2013 Nature Springer. (d) Gradient magnetometry signal associated with the fully polarized twisted bilayer graphene device. Adapted with permission from ref (255). Copyright 2021 AAAS. (e–g) Magnetic images obtained with scanning NV magnetometry. (e) Magnetization image of bilayer/trilayer CrI₃ adapted with permissions from ref (21). Copyright 2019 AAAS. (f) The magnetic domains in bilayer CrBr₃ adapted with permission under a Creative Commons CC BY license from ref (256). Copyright 2021 Springer Nature. (g) Stray field map of a magnetic skyrmion in the CoFeB system adapted with permission under a Creative Commons CC BY license from ref (257). Copyright 2018 Springer Nature.

Figure 26

(a) Crystal structure of Cr₂Ge₂Te₆. (b) Temperature dependence of magnetization for Cr₂Ge₂Te₆ measured in H = 1 kOe. Inset: Field dependence of magnetization at T = 2 K. (c) 2D Ising model plot of isotherms around T_c. (d) Modified Arrott plot of M^1/βversus (H/M)^1/γ with β = 0.194 and γ = 1.36. The straight line is the linear fit of isotherm at T = 62.5 K. (e) Temperature dependence of the spontaneous magnetization M_s (left) and the inverse initial susceptibility (right) with solid fitting curves. (f) Kouvel–Fisher plots of (left) and (right) with solid fitting curves. (g) Isotherm MversusH plot collected at T_c = 62.7 K. Inset: The same plot in log–log scale with a solid fitting curve. (h) Scaling plots of renormalized magnetization mversus renormalized field h below and above T_c for Cr₂Ge₂Te₆. Inset: The rescaling of the M(H) curves by MH^–1/δversus εH^–1/(βδ). All panels adapted with permission from ref (279). Copyright 2017 American Physical Society.

Figure 27

Temperature dependence of (a) |ΔS_M| and (b) ΔC_p in different fields with H ∥ c; Field dependence of parameters from |ΔS_M(T)| with the fitted curves: (c) , (d) δT_fwhm, and (e) RCPversusH with the fitted curves; (f) Modified Arrott plot based on the obtained critical exponents. Scaling of the |ΔS_M(T, H)| curves: (g) normalized ΔS_M(T, H) as a function of θ (inset gives T_r1 and T_r2 as a function of H); (h) – ΔS_M/H^(1−α)/Δversus ε/H^1/Δ. All panels adapted with permission from ref (280). Copyright 1998 American Physical Society.

Figure 28

(a) Schematics illustrating the combination of two structural motifs, i.e., that of transition-metal dichalcogenides TMC₂ (TM, transition metal; C, chalcogen) together with that of body-centered cubic iron to form Fe-rich vdW coupled ferromagnets of composition Fe_xGeC₂. (b) Three stable structures within the Fe_nGeTe₂ series with the values n = 3, 4, and 5 which were previously identified through ab initio calculations. Fe–Fe dumbbells are the common structural units that form multiple-layer Fe-rich slabs stacked through vdW-like coupling. As n increases, the number of nearest Fe neighbors per Fe atom gradually increases, which is expected to enhance the pair exchange interaction and, thus, T_c. All panels are adapted with permission under a Creative Commons CC BY-NC 4.0 license from ref (88). Copyright 2020 AAAS.

Figure 29

(a) Magnetic susceptibility χ as a function of the temperature T for a Fe₃GeTe₂ single-crystal measured under zero field-cooled (black markers) and field cooled (red markers) conditions. Open markers correspond to the inverse of χ where the blue line is a linear fit. The Curie temperature exceeds 200 K. Adapted with permission from ref (327). Copyright 2017 American Physical Society. (b) χ as a function of T for a Fe₄GeTe₂ single-crystal and for fields along the c-axis (solid green makers) and the ab-plane (open green markers). Solid black markers depict both the resistivity and its derivative indicating a T_c of ∼270 K with another spin-reorientation transition above 100 K. Adapted with permission under a Creative Commons CC BY-NC 4.0 license from ref (88). Copyright 2020 AAAS. (c) Magnetization as a function of the temperature for Fe₅GeTe₂ for fields along the c-axis (open markers) and the ab-plane (solid markers) and also polycrystalline material (gray line). It displays a T_c in excess of 280 K. Adapted with permission from ref (87). Copyright 2019 American Chemical Society.

Figure 30

(a) Atomic structure of monolayer Fe₃GeTe₂. The left panel shows the view along [001]; the right panel shows the view along [010]. Bulk Fe₃GeTe₂ is a layered crystal with an interlayer vdW gap of 2.95 Å. Fe_I and Fe_II represent the two inequivalent Fe sites in the +3 and +2 oxidation states, respectively. (b) Optical image of typical few-layer flakes exfoliated on an Al₂O₃ thin film. (c) Atomic force microscopy image of the area marked by the square in (b). Mono- and few-layer flakes are clearly visible. Scale bar, 2 μm. (d) Cross-sectional profile of the Fe₃GeTe₂ flakes along the white line in (c). The steps are 0.8 nm in height, or consistent with the thickness (0.8 nm) of monolayer (1L) Fe₃GeTe₂. (e) Normalized remanent anomalous Hall resistance as a function of temperature obtained from Fe₃GeTe₂ thin-flake samples with varying numbers of layers. Arrows mark the FM transition temperature T_c. (f) Phase diagram of Fe₃GeTe₂ as layer number and temperature are varied. T_c values are determined from anomalous Hall effect, Arrott plots and RMCD are displayed in blue, red and magenta, respectively. (g) Remanent RMCD signal as a function of temperature for a sequence of selected few-layer flakes (1 L, monolayer; 2 L, bilayer; 3 L, trilayer; 4 L, four layers; 5 L, five layer). The solid lines are least-squares criticality fits of the form α(1 – T/T_c)^β. Inset: derived values of the exponent β plotted as a function of thickness. (h) Thickness-temperature phase diagram. PM denotes the region in which the flake is paramagnetic, FM1 that in which it is FM with a single domain and FM2 that in which the flake exhibits labyrinthine or stripe domains. The transition temperatures, T_c, T_c1, and T_c2, are based on the temperature-dependent RMCD or anomalous Hall effect measurements for each flake thickness. The red dashed line denotes the critical thickness at which a dimensional crossover occurs. All panels are adapted with permission from ref (12). Copyright 2018 Springer Nature.

Figure 31

Principle of a μSR experiment. (a) Overview of the experimental setup. Spin polarized muons with spin S_μ antiparallel to the momentum p_μ are implanted in the sample placed between the forward (F) and the backward (B) positron detectors. A clock is started at the time the muon goes through the muon detector (M) and is stopped as soon as the decay positron is detected in the detectors F or B. Adapted with permission from ref (348). Copyright Swiss Physical Society. (b) The number of detected positrons N_F and N_B as a function of time for the forward and backward detector, respectively. Reproduced with permissions from ref (349). Copyright University of Zurich. (c) The so-called asymmetry (or μSR) signal is obtained by essentially building the difference between N_F and N_B (eq 2). All panels are adapted with permission under a Creative Common CC BY license from ref (350). Copyright 2019 MDPI.

Figure 32

(a-c) Schematic illustration of the magnetically homogeneous (i.e., full volume magnetic) (d), inhomogeneous (full volume magnetic, but with domains) and phase separated (i.e., part of the volume magnetic and part paramagnetic) polycrystalline samples and the corresponding μSR spectra. The 1/3 nonoscillating μSR signal fraction originates from the spatial averaging in powder samples where 1/3 of the magnetic field components are parallel to the muon spin and do not cause muon spin precession. (d, e) Isotropic Gaussian field distribution for polycrystalline sample. Panels (a–c) adapted with permission under a Creative Common CC BY license from ref (350). Copyright 2019 MDPI.

Figure 33

Types of intrinsic disorder in TMDs: (a) vacancy, (b) antisite, (c) substitution. Panels (a–c) adapted with permission from ref (380). Copyright 2019 Springer Nature. (d) Schematic diagram of the phase incorporation strategy to achieve ferromagnetism of 2H-MoS₂ nanosheets. Adapted with permission from ref (381). Copyright 2015 American Chemical Society.

Figure 34

(a) ZF μSR time spectra for the single crystal samples of T_d-MoTe₂ and 2H-MoTe₂ recorded at T = 5 K. Adapted with permission under a Creative Common CC BY license from ref (350). Copyright 2019 MDPI. (b) Temperature dependence of the internal field H_int of 2H-MoTe₂, 2H-MoSe₂ as a function of temperature. Adapted with permission under a Creative Common CC BY-NC 4.0 license from refs (382). Copyright 2019 AAAS.

Figure 35

(a) Large-scale atomic-resolution STM topography (20 nm) of the 2H-MoTe₂ surface. The image reveals an approximately uniform density of two types of defects over the entire surface. (b) DFT+U-optimized geometry for Mo_sub defect. (c) DFT+U-optimized geometry of the Mo vacancy Mo_vac. (d) Magnetization density (0.001 electrons/bohr³) on the top surface of bulk 2H-MoTe₂ in AF configuration. Spin-up and spin-down states are shown in faint blue and orange isosurfaces, respectively. Note that spins also couple antiferromagnetically at the local level between the Mo impurity and the nearest Mo atoms. All panels are adapted with permission under a Creative Common CC BY-NC 4.0 license from ref (382). Copyright 2019 AAAS.

Figure 36

Calculated magnetization of the antisite defect. (a) Occupation number versus rigid potential shifts α for antisite defects for the bare, noninteracting potential χ₀ and the interacting potential χ. From the angular coefficients of both curves we can extract the optimum U_LR for our system, U_LR = – χ^–1. (b) Variation of the local magnetization at the defect antisite versus U. At U = 0, no magnetic moments are observed as the defect shows a symmetric configuration at the Mo–Mo bonds. At U > 0.5 eV, this symmetry is broken and the defect develops an appreciable magnetic moment that increases with U as a result of the increased localization of the bands. All panels are adapted with permission under a Creative Common CC BY-NC license from ref (382). Copyright 2019 AAAS.

Figure 37

Temperature dependence of the magnetic volume fraction for CrI₃. Adapted with permission under a Creative Commons CC BY license from ref (383). Copyright 2021 Springer Nature.

Figure 38

(a) Temperature dependence of the magnetic volume fraction for VI₃, determined from weak transverse field μSR experiments. (b) Zero-field μSR spectra, recorded at T = 5 and 60 K. (c) The temperature dependence of the internal fields H_int for VI₃. Original figure, no permissions needed.

Figure 39

Voltage control of the magnetic properties of CrI₃, CGT and FGT. (a) Top: Normalized magnetization measured by MCD as a function of the applied electric field (trace and retrace) at 4 K and fixed magnetic field (+0.44 T for top panel and −0.44 T for bottom panel), showing the electrical switching of the magnetic order in bilayer CrI₃. The insets represent the corresponding magnetic states. Adapted with permission from ref (8). Copyright 2018 Springer Nature. (b) Uniaxial magnetic anisotropy field of multilayer CGT as a function of temperature at different gate voltages and in the pristine case. Inset: The dependence of T_C on gate voltage. Adapted with permission from ref (86). Copyright 2020 Springer Nature. (c) T_C of a trilayer FGT as a function of gate voltage. (d) H_C of a trilayer FGT as a function of gate voltage at 10 K. Panels (c) and (d) are adapted with permission from ref (12). Copyright 2018 Springer Nature.

Figure 40

Current-induced magnetization switching of a FGT 2D magnet. (a) Schematic illustration of the current-induced switching of a FGT nanoflake by the spin current generated by SHE in the Pt layer deposited on FGT and injected into FGT to produce a SOT.J_x is the current in Pt generating a downward spin current by SHE, H₀ is the in-plane field tilting the magnetization M from its out-of-plane orientation at zero field, H_DL is the damping-like (DL) component of the effective field expressing the action of the spin transfer torque. (b) Current-induced switching detected by the change of sign of the transverse voltage V_H in panel (a) (R_H ∼ V_H/J_x). Panels (a) and (b) are adapted with permission from ref (442). Copyright 2019 American Chemical Society. (c) Schematic view of the CrI₃/TaSe₂ vdW heterostructure consisting of an insulating AF bilayer of CrI₃ and a nonmagnetic metallic monolayer TMD TaSe₂. The SOT on the magnetization of CrI₃ is due to the charge-to-spin Edelstein conversion of the current flowing along the CrI₃/TaSe₂ interface. The resulting switching of m₁ is indicated by arrows. Adapted with permission from ref (444). Copyright 2020 American Chemical Society.

Figure 41

Structure and magnetic properties of Bi₂Te₃/FGT heterostructures. (a) Structure of Bi₂Te₃/FGT. (b) Anomalous Hall resistance R_xy as a function of temperature for FGT and Bi₂Te₃/FGT heterostructures. (c) Magnetic phase diagram of pure FGT and Bi₂Te₃/FGT heterostructures versus FGT layer thickness and temperature (FM for ferromagnetic, PM for paramagnetic). All panels are adapted with permission from ref (90). Copyright 2020 American Chemical Society.

Figure 42

Spin Seebeck effect and magnon transport with 2D magnets. (a) Crystal structure of CGT and CST. (b) Schematic of the longitudinal SSE measurements in CST/Pt or CGT/Pt bilayers. H denotes the external magnetic field and ΔT (∇T) the temperature difference (gradient). (c, d) Normalized SSE voltage S = (V/∇T) (L_z/L_y) as a function of H in the (c) CST/Pt and (d) CGT/Pt bilayers at selected temperatures. Panels (a–d) are adapted with permission from ref (467). Copyright 2019 American Physical Society. (e) Schematic of the magnon generation, transport, and detection in MnPS₃. (f) Optical image of the device with the MnPS₃ flake and Pt electrodes, including the measurement configuration of the nonlocal SSE. (g) Normalized nonlocal signal as a function of distance (d) for selected temperatures in a 16 nm-thick MnPS₃ flake. The solid lines represent the best-fitting results based on a diffusion equation. (h) Magnon diffusion length as a function of MnPS₃ thickness (t) for selected temperatures. Panels (e–h) are adapted with permission under a Creative Commons CC BY 4.0 license from ref (468). Copyright 2019 American Physical Society.

Figure 43

Skyrmions in 2D magnets. (a) Spin structure of a Néel skyrmion. (b) Pt/Co interface in which the absence of inversion symmetry generates DMI, H_DMI = −(S₁ × S₂)·D₁₂. (c) Side view of the crystal structure of the Janus TMD MnSTe in which the absence of inversion symmetry generates DMI. (d) DMI strength calculated for the Janus TMD MnSeTe, MnSTe, and MnSSe. (e) Skyrmions in MnSeTe (T = 10 K in applied field of 0.3 T) from DFT calculation of DMI and Monte Carlo simulations. Panels (c–e) are adapted with permission from ref (480). Copyright 2020 American Physical Society. (f) Side view of the crystal structure of WTe₂ on FGT. (g) LTEM image of Néel skyrmion lattice at 180 K under 510 Oe in the sample 2L WTe₂/40L FGT (L = layer) at tilt angle 30° and under focus. Scale bar: 500 nm. Panels (f) and (g) are adapted with permission under a Creative Commons CC BY license from ref (27). Copyright 2020 Springer Nature. (h) TEM image of Néel skyrmion lattice in an oxidized FGT flake (about 50 μm thick) at 160 K, tilt angle 20° and over focus. Adapted with permission from ref (26). Copyright 2021 American Physical Society. (i) Magnetization maps derived from analysis of LTEM image for Bloch bubbles in a CGT flake at 17 K in a field of 11.7 mT. Adapted with permission from ref (481). Copyright 2019 American Chemical Society.

Figure 44

Toward MRAM with 2D magnets. (a) Schematic of STT-MRAM with 3D materials: information coded by the relative orientations of the magnetization of the two magnetic layers (green) separated by an insulating MgO layer (red), writing by current-induced STT and reading by TMR. Adapted with permission from ref (491). Copyright 2017 American Physical Society. (b) Schematic of the TMR device based on a FGT/hBN/FGT vertical stack used for the results shown in panels (c–e). (c) TMR measurement in a FGT/hBN/FGT vertical stack at 4.2 K. The swapped magnetic field is out of plane. (d, e) Magnified regions of the TMR measurement around the field range of antiparallel configuration (upper panels) and variation of the AHE resistance of the top (blue) and bottom (green) FGT electrodes in the same field range. Blue and green arrows indicate the successive orientations of the magnetizations. Panels (b–e) are adapted with permission from ref (109). Copyright 2018 American Chemical Society. (f) Schematic of the device combining a gate-controlled spin-flip transition in bilayer CrI₃ and spin filtering in the tunnel junction. Arrows indicate the magnetic orientation of each layer. (g) Tunnel conductance of the device illustrated in panel f as a function of gate voltage (sweeping back and forth) under a constant magnetic field (0.76 T). The measured tunnel conductance changes when the magnetic order of bilayer CrI₃ is switched by the gate. Panels (f) and (g) are adapted with permission from ref (492). Copyright 2019 Springer Nature.

Figure 45

2D magnet-based SOT-MRAMs. (a) Top: Schematic of a SOT-MRAM based on 3D materials in which an electrical current in the heavy metal (Ta) of the bottom electrode generates by SHE the vertical spin current injected in the bottom FeCoB layer. This injection of spin current switches the magnetization of FeCoB by SOT (writing). The state of the memory is detected by the TMR of the FeCoB/MgO/FeCoB MTJ (reading). Bottom: Detection by TMR of the SOT-induced switching of the magnetization of the bottom FeCoB layer in the device of the schematic. Adapted with permission from ref (507). Copyright 2014 AIP Publishing. (b) Top: Schematic of a bilayer for SOT-MRAM in which the orientation of the out-of-plane magnetization of a FGT layer codes the information and is switched by the SOT generated by the SHE of the Pt layer. As shown in the bottom part of the figure, the switching is detected by the AHE resistance R_xy derived from the voltage between transverse contacts. Adapted with permission under a Creative Commons CC BY-NC 4.0 license from ref (443). Copyright 2019 AAAS. (c) Image of a heterostructure for SOT-MRAM in which the magnetic state of a CGT layer can be switched by the SHE of a Ta layer. (d) Comparison of the current densities and in-plane fields required for SOT switching in devices based on 3D magnetic layers (CoFeB, MnGa, thulium iron garnet (TmIG)) and 2D magnets (FGT, CGT). The best results so far are for Ta/CGT. Panels (c) and (d) are adapted with permission from ref (446). Copyright 2020 John Wiley and Sons.

Figure 46

Devices based on current-induced motion of skyrmions in 2D magnets. (a) Skyrmion motion induced by current pulses in a FGT track. Each STXM image is acquired after injecting five unipolar current pulses of 50 ns. Two individual Néel skyrmions are outlined in colored circled for clarity. The diameter of the skyrmions is about 200 nm, their velocity is around 1 m/s for a current density of 1.4 × 10¹¹ A/m² and the width of the track is 50 μm. Adapted with permission from ref (26). Copyright 2021 American Physical Society. (b) Schematic of racetrack memory storing data by aligning skyrmions like beads on an abacus and displacing them by current-induced SOT from write head to read head. Adapted with permission from ref (476). Copyright 2018 AIP Publishing. (c) Proposal of skyrmion-based racetrack memory based on the SOT-induced motion of antiferromagnetically-coupled skyrmions in two layers coupled by AF interactions. The left inset is a schematic of an antiferromagnetic (AFM or AF)-coupled nanotrack and the right inset represents AF-coupled skyrmions in Co/Ru/Co trilayers. As the AF-coupled skyrmions have the same chirality but opposite polarities, their motion has the advantage of being along the current direction (no Skyrmion Hall effect^,,). Note that the racetrack memory of the schematic includes not only injector and detector but also an update/delete/insert. Adapted with permission from ref (513). Copyright 2018 IEEE. (d) AF-coupled CrI₃ layers in a CrI₃ bilayer. Adapted with permission from ref (9). Copyright 2018 Springer Nature.

Figure 47

(a) Magnetic structure of Co₃Sn₂S₂, showing a FM ground state with spins on Co atoms aligned along the c-axis. (b) Kagome lattice structure of the Co₃Sn layer. (c) Topographic image of the CoSn surface. (d) A zoom-in image of the CoSn surface (left) that shows similar morphology with the FeSn surface (right) in Fe₃Sn₂. The inset illustrates the possible atomic assignment of the kagome lattice. Panels (a–d) are adapted with permission under a Creative Commons CC BY 4.0 license from ref (514). Copyright 2020 Springer Nature. (e) The temperature dependence of the intrinsic anomalous hall conductivity in Co₃Sn₂S₂. (f) Left: linear band crossings form a nodal ring in the mirror plane. Right: Spin–orbit coupling breaks the nodal ring band structure into opened gaps and Weyl nodes. The Weyl nodes are located just 60 meV above the Fermi level, whereas the gapped nodal lines are distributed around the Fermi level. Panels (e) and (f) are adapted with permission from ref (515). Copyright 2018 Springer Nature.

Figure 48

(a) Upper panel: Illustration of the magnetization-polarized Zeeman effect for Co₃Sn₂S₂. Lower panel: illustration of the large negative orbital magnetism of the flat band in the kagome lattice. (b) Upper panel: Orbital magnetism for the flat band calculated from first principles. The magnetic moment (red arrows) is plotted along the flat band. The red bar marks the units of the magnetic moment value. Lower panel: Orbital magnetism from the magnetic kagome lattice model. The magnetic moment (red arrows, arbitrary unit) is plotted along the flat band. All panels adapted with permission from ref (516). Copyright 2019 Springer Nature.

Figure 49

(a) The temperature dependence of the relative volume fractions of the two magnetically ordered regions. Arrows mark the critical temperatures T_C1 and T_C2 for FM and AF (or AFM) components, respectively as well as the transition temperature , below which only FM component is observed. (b) Spin structures of Co₃Sn₂S₂, i.e., the FM and the in-plane AF (or AFM) structures. (c) The correlation plot of anomalous hall conductivity versus FM fraction. (d) Calculated AHC for out-of-plane FM and in-plane AF structures. The inset shows the calculated Berry curvature distribution in the BZ at the FM phase. All panels are adapted with permission under a Creative Commons CC BY 4.0 license from ref (514). Copyright 2020 Springer Nature.

Figure 50

(a) Calculated fractions of the ferromagnetism F and antiferromagnetism (1 – F) in Co₃Sn₂S₂ and (b) averaged Chern number as a function of the in-plane AF correlation J_xy, taking into account fluctuations in the Hund’s coupling. All panels are adapted with permission under a Creative Commons CC BY 4.0 license from ref (535). Copyright 2020 American Physical Society.

Figure 51

Mechanical exfoliation methods. (A) Schematic of the Al₂O₃ film-assisted mechanical exfoliation. The strong adhesion between the crystal and the Al₂O₃ film makes it possible to exfoliate layered crystals that are otherwise difficult to cleave from SiO₂ surfaces using conventional methods. Adapted with permission from ref (12). Copyright 2018 Springer Nature. (B) Schematic of the Au-assisted exfoliation process. First, a thin layer of Au is deposited onto a substrate, then a freshly cleaved bulk crystal is placed on the Au layer. The Au is then removed with a KI/I₂ aqueous solution etchant. Adapted with permission under a Creative Commons CC BY license from ref (566). Copyright 2020 Springer Nature.

Figure 52

Identification of exfoliated layer numbers. (A) Optical contrast map of a representative CrI₃ flake (left) and the corresponding optical contrast per number of layers (right). Adapted with permission from ref (5). Copyright 2017 Springer Nature. (B) Left: Optical image of few-layer flakes of MnBi₂Te₄ exfoliated onto Al₂O₃. The corresponding layer number is labeled on selected flakes. The scale bar is 20 μm. Right: MnBi₂Te₄ transmittance versus layer number. Adapted with permission from ref (111). Copyright 2020 AAAS. (C) Optical image of few-layer Fe₃GeTe₂ flakes exfoliated onto Al₂O₃. (D) AFM image of the area in (C) marked by a solid black square. The scale bar is 2 μm. (E) Height profile plotted versus length along the white line in (D). Panels (C–E) are adapted with permission from ref (12). Copyright 2018 Springer Nature.

Figure 53

Scheme for the one-step synthesis and vapor transport of CrX₃ (X = Br, I) micro- and nanosheets directly on YSZ substrates shown by the example of CrBr₃. Prior to the CVT process for deposition of the respective nanolayers the introduction of chromium powder and bromine (Br₂ in small sealed capillaries) lead to the formation of CrBr₃ (solid) (Cr_(s) + 1,5 Br_2(l) → CrBr_3(s)) and gaseous CrBr_n(g) (n = 2, 3, 4) at T₂. By application of a temperature gradient (T₂ → T₁) chemical vapor transport is achieved for deposition of CrBr₃ micro- and nanosheets on YSZ substrates directly. The reaction course is similar for the formation of CrI₃, while in contrast to this scheme CrCl₃ is utilized as presynthesized compound, that is not introduced by mixture of the elements. Reproduced with permission from ref (600). Copyright 2019 John Wiley and Sons.

Figure 54

Crystal growth by vapor transports of CrCl₃ on YSZ substrates. (a) Optical microscopy of CrCl₃ micro- and nanocrystals at YSZ substrate. (b) Optical microscope image of YSZ substrate surface with CrCl₃ sheets with respective thicknesses in dotted boxes. (c) Distribution of thicknesses of CrCl₃ structures on a YSZ substrate after CVT (yellow), after one time of exfoliation (green) and after three times of exfoliation (purple). (d) AFM measurement the height profile of a CrCl₃ nanosheet. (e) Corresponding AFM image of measurement of (d) the white arrow is indicating the measurement. (f) AFM measurement of the height profile a CrCl₃ ultrathin sheet (red line) and monolayer (purple line) after three repeats of exfoliation. (g) Corresponding AFM image of measurement of (f) the white line is indicating the monolayer AFM measurement. All panels are reproduced with permission from ref (600). Copyright 2019 John Wiley and Sons.

Figure 55

(A, B) RHEED patterns with indicated diffraction orders of (A) the bare HOPG substrate and (B) the MBE-grown CrBr₃ film. (C, D) STM images of (C) the CrBr₃ monolayer with (D) bilayer islands. The scan parameters were as follows: V_b = 1.1 V, I = 100 pA, T = 5 K for (C) and V_b = 1.5 V, I = 100 pA, T = 5 K for (D). (E) Atomically resolved image of a monolayer CrBr₃ with an overlaid atomic structure. The scan parameters were as follows: V_b = 1.5 V, I = 500 pA, T = 5 K. The lattice constants were determined to be 6.3 Å for the primitive vectors a and b, consistent with the bulk values. (F) Illustrations of the top and side views of the monolayer CrBr₃ atomic structure. The Cr atoms form a honeycomb lattice sandwiched by Br atoms. Within the Cr honeycomb lattice, the top and bottom surfaces of Br atoms form single triangles but with opposite orientation, indicated by solid and dotted green lines, respectively. (G) AFM image of monolayer CrBr₃ with partial coverage. A line-cut profile across the monolayer and bare substrate is shown with a monolayer height of 6.5 Å. All panels are adapted with permission from ref (20). Copyright 2019 AAAS.

Figure 56

(A, B) The structure and ground states of iron-deficient Fe_3–xGeTe₂. (A) Side view of stoichiometric Fe₃GeTe₂. Fe_I (red) and Fe_II (silver) are two inequivalent Fe sites with +3 and +2 formal charges, respectively. The Fe_I–Fe_I interactions are mostly responsible for interlayer AF ordering while Fe_I-Fe_II and Fe_II–Fe_II couplings are FM. With Fe defects or doping, Fe_I–Fe_II and Fe_II–Fe_II become dominant and push interlayer ordering into FM. (B) The calculated energy differences (ΔE) between interlayer AF and FM phases as a function of hole concentration. For ΔE > 0, FM is favored (between 0.2 and 0.6 holes per formula unit). Panels (A) and (B) are adapted with permission from ref (615). Copyright 2020 American Chemical Society. (C) The layered crystal structure of MBT. The red arrow indicates the Mn sites in which antisites have been identified. (D) STM of the surface of a cleaved MBT crystal; white spots show the presence of multiple antisite point defects. Panels (C) and (D) are adapted with permission from ref (616). Copyright 2020 American Chemical Society. (E) Crystal structure representation of the (MnBi₂Te₄)_m(Bi₂Te₃)_n series, ranging from AF to FM with various compositions derived from codepositional MBE. Here, we see that with the addition of Bi₂Te₃ layers, the MnBi₂Te₄ layers cannot be coupled together and the material becomes less AF. This is a prime example of an off-stoichiometry defect. Panel (E) adapted with permission from ref (617). Copyright 2020 AIP Publishing.

Figure 57

Common experimental techniques for measuring the Young’s modulus and strength of 2D materials. (a) Nanoindentation based on AFM. Adapted with permission under a Creative Commons CC BY license from ref (634). Copyright 2019 John Wiley and sons. (b) Bulge test involving interferometry and Raman spectroscopy. Adapted with permission from ref (635). Copyright 2017 American Physical Society. (c) Tensile testing push-to-pull micromechanical device controlled by an external pico-indenter in SEM, and the yellow arrow shows the loading direction. (d) Diagram showing the enlarged pink rectangle area in (c) with a suspended graphene sample after cutting by focused ion beam.^, Copyright 2009 Royal Society of Chemistry, copyright 2019 American Physical Society, copyright 2019 John Wiley and Sons, copyright 2017 American Physical Society, copyright 2020 Springer Nature, copyright 2014 Springer Nature. Panels (c) and (d) are adapted with permission under a Creative Commons CC BY 4.0 license from ref (636). Copyright 2020 Springer Nature.

Figure 58

Mechanical properties of CrCl₃ and CrI₃. (a) Optical microscopy images of 2L and few layers CrCl₃ and Crl₃ suspended over microwells (600 nm in diameter) on a SiO₂/Si substrate. (b) Load–displacement curves and the corresponding fittings of 2L and 7L CrCl₃ and CrI₃. (c) Volumetric Young’s modulus and breaking strength of 2–10L CrCl₃ and CrI₃, along with dashed lines that show the linear fits. (d) Demonstration of the good plasticity of bulk CrCl₃ and CrI₃ crystals via folding them into rings. (e) Deformability factor versus Young modulus, where I, II, and III correspond to plastic-flexible, potentially deformable and brittle-rigid regions, respectively. The experiential results of CrCl₃ and CrI₃ are shown as red-filled circles, and the other layered vdW materials are shown as green filled circles. (f) Deformability factor versus bandgap for the same materials as in (e), and the materials that may show exceptional plastic behavior are shown in the dashed line encircled green area. All panels are adapted with permission from ref (659). Copyright 2021 American Chemical Society.

Figure 59

Modulus-strength graph. The Young’s modulus and fracture strength of 2D magnets are compared with those of conventional bulk materials and other 2D materials. Adapted with permission from ref (659). Copyright 2021 American Chemical Society.

Figure 60

Mechanical properties of ternary compounds. (a) Changes in the magnetic configurations of various MPX₃ (M = Mn, Fe, Ni; X = S, Se) compounds at zero carrier density under in-plane biaxial compressive and expansive strains. Adapted with permission from ref (684). Copyright 2016 American Physical Society. (b) Magnetostriction in 2L CrI₃ and (c) 6L CrI₃ resonators under an out-of-plane magnetic field (μ₀H_⊥) and in (d) 6L CrI₃ resonator under an in-plane magnetic field (μ₀H_∥). The resonance frequency (b–d) and MCD (b, c) of the membranes as a function of the magnetic field, where the red (blue) lines correspond to the measurement for the positive (negative) sweeping direction of the field. Panels (b–d) are adapted with permission from ref (685). Copyright 2020 Springer Nature.

Figure 61

Magnetic order and spin excitations in vdW magnetic materials with tunable fundamental spin Hamiltonian’s and structural parameters probed by neutron scattering methods. Adapted with permission from ref (84). Copyright 2018 Springer Nature.

Figure 62

Magnetic structures of (a) FePS₃. Adapted with permission from ref (724). Copyright 2016 American Physical Society. (b) MnPS₃. adapted with permission from ref (318). Copyright 2006 American Physical Society. (c) NiPS₃. adapted with permission from ref (317). Copyright 2015 American Physical Society. (d) CoPS₃. adapted with permission from ref (729). Copyright 2017 IOP Publishing. All of these materials have honeycomb lattice structure and are antiferromagnetically ordered.

Figure 63

(a) A spin gap at Dirac point in CrI₃ suggests that spin excitations in this system can have chiral and topological edge mode. Adapted with permission under a Creative Commons CC BY 4.0 license from ref (744). Copyright 2018 American Physical Society. (b) The FM phase transition in CrI₃ is weakly first-order and controlled by spin gap. Adapted with permission from ref (745). Copyright 2020 American Physical Society.

Figure 64

(a) The phase diagram of α-RuCl₃ adapted with permission from ref (752). Copyright 2018 Springer Nature. At zero field, the system forms a zigzag magnetic structure as shown in the left inset. For in-plane magnetic field between 7 and 9 T, the system is believed to be in Kitaev QSL state. For fields above 9 T, the system becomes non topological from thermal transport measurements. (b) Wave vector/energy dependence of spin excitation of α-RuCl₃ at 4 and 8 T, adapted with permission under a Creative Commons CC BY 4.0 from ref (753). Copyright 2018 Springer Nature. One can clearly see spin waves stemming from zigzag ordered wave vector (0.5, 0, 0) at 4 T. The scattering centered around Γ points is believed to arise from fractionalized excitations of a Kitaev QSL, which is enhanced upon suppression of spin waves with a 8 T in-plane magnetic field.

Figure 65

(a) The 2D kagome lattice structure. (b) Calculated electronic dispersion and flat band. (c, d, e) Crystal structures of T₃X, T₃X₂, and TX, respectively, where T = Fe, Mn, Co, and X = Ge, Sn. All panels are adapted with permission from ref (766). Copyright 2019 Springer Nature.

Figure 66

(a) Red arrows indicate in-plane spin directions at zero field. Black arrows indicate the moment direction in a c-axis aligned field. (b) Spin-wave branches for one acoustic and two optical bands. The middle and top green arrows indicate Dirac points and band top, respectively. (c) Nearest-neighbor DMI direction. (d) DMI induced spin gap at Dirac points. (e) Neutron scattering measured spin waves energy/moment map at (a) 0 T, (b) 2 T, (c) 7 T; (d) Calculated neutron spectra for 7 T. All panels are adapted with permission from ref (767). Copyright 2015 American Physical Society.

Figure 67

(a) A-type AF structure of FeSn in 2D kagome lattice structure, adapted with permission from ref (779). Copyright 2019 American Physical Society. (b) The antichiral noncollinear structure of Mn₃Ge, adapted with permission from ref (673). Copyright 2020 American Physical Society. (c, d) Complicated noncollinear magnetic structures of barlowite with proximate 2D kagome lattice structure, adapted with permission under a Creative Commons CC BY 4.0 license from ref (780). Copyright 2020 Springer Nature.

Figure 68

(a) Top: Schematic illustrating IETS mechanism. Bottom: Color plot of IETS spectra (|(d²I)/(dV²)| versus V) taken on bilayer CrI₃ as a function of out-of-plane magnetic field (easy axis) shows two dispersing magnon modes. Bottom panel adapted with permission from ref (120). Copyright 2019 National Academy of Sciences of the United States of America. (b) Top: Spin-wave calculations of magnon dispersion in 2D CrI₃ from anisotropic Heisenberg model with nearest-neighbor interactions. Adapted with permission from 217. Copyright 2018 Springer Nature. Magnon energies are shown at high-symmetry Γ and M points as a function of nearest-neighbor exchange energy J and anisotropy α. Zoom-in of the acoustic branch at the Γ point shows an energy gap for α > 1. (c) Extracted J and α values for all three 2D chromium trihalides from IETS measurements.

Figure 69

(a) Top: Experimental Raman spectra (left) and theoretical calculations (right) of magnon modes in bulk CrI₃ as a function of out-of-plane magnetic field (easy axis). Above B_c ∼ 2 T, a single mode is observed as expected for a fully spin-polarized state for all layers. Below B_c, two additional modes are seen that disperse oppositely with field, corresponding to the layer-AF order on surface layers. Bottom: Schematic of spins in bulk CrI₃ above and below B_c. Adapted with permission under a Creative Commons CC BY 4.0 license from ref (218). Copyright 2020 American Physical Society. (b) Field-dependent Raman spectra of bilayer CrI₃. Only layer-AF magnons modes are obtained. Adapted with permission from ref (794). Copyright 2020 Springer Nature.

Figure 70

Process flow for vdW heterostructure fabrication. Schematic of the dry-polymer-transfer process used to fully encapsulate a 2D flake with hBN (here, it is graphene). This process relies only on the vdW interactions between hBN and other 2D flakes and can be used to pick up and transfer a number of different 2D materials. Reproduced with permission from ref (577). Copyright 2013 AAAS.

Figure 71

Contact methods for exfoliated vdW magnets. (A) Schematic of a Cr₂Ge₂Te₆ flake top contacted with metal electrodes under an ion-liquid gate. Adapted with permission from ref (86). Copyright 2020 Springer Nature. (B, C) Optical image of a 5.8 nm-thick Fe₃GeTe₂ (B) and a 28 nm-thick Fe₅GeTe₂ (C) flake contacted from the bottom with prepatterned metal electrodes. In (B) the red dashed line is the Fe₃GeTe₂ and the yellow dashed line is the hBN. The red scale bar is 10 μm. In (C) the scale bar is 20 μm. Panel (B) was adapted with permission from ref (123). Copyright 2018 Springer Nature. Panel (C) was adapted with permission from ref (87). Copyright 2019 American Chemical Society. (D–G) Schematics (D, F) and corresponding false-colored optical images (E, G) of CrI₃ flakes contacted by graphene electrodes for lateral (D, E) and tunneling (F, G) transport measurements. In (E) and (G), the scale bars are both 5 μm. Panels (D–G) are reproduced with permission from ref (15). Copyright 2018 Springer Nature.

Figure 72

Heterostructures fabricated from 2D vdW magnets. (A) Schematic of a Cr₂Ge₂Te₆ flake fully encapsulated with hBN and contacted by graphene electrodes. Reproduced with permission from ref (10). Copyright 2018 Springer Nature. (B) Schematic (left) and corresponding optical image (right) of a MTJ fabricated from Fe₃GeTe₂ electrodes separated by an hBN barrier. In the right panel, the dotted lines outline the edges of the two Fe₃GeTe₂ flakes. The scale bar is 5 μm. Reproduced with permission from ref (109). Copyright 2018 American Chemical Society. (C) Optical image (left) and corresponding cartoon (right) of a spin-filter MTJ utilizing CrI₃ as the tunnel barrier between graphene electrodes. Reproduced with permission from ref (14). Copyright 2018 AAAS. (D) Schematic (top) and a false-colored optical image (bottom) of a spin-field-effect transistor fabricated from 4-layer CrI₃. Graphene acts as both transistor electrodes and local electrostatic gates. The scale bar is 5 μm. Reproduced with permission from ref (504). Copyright 2019 American Chemical Society. (E, F) Schematic (E) and false-colored optical image (F) of a heterostructure proximitizing CrI₃ with WSe₂. The scale bar in (F) is 5 μm. Panels (E, F) are reproduced with permission under Creative Commons CC BY-NC 4.0 license from ref (206). Copyright 2017 AAAS.

Figure 73

2D magnetism controlled by twistronics and stacking order. (A) Variation of stacking order in small twist angle θ monolayer–bilayer graphene (tMBG). (B) Transverse resistance R_yx map measured versus total carrier density n and the magnetic field B at T = 6.4 K in a tMBG device with θ = 1.25° near ν = 3 orbital magnetic state presents a magnetization reversal driven by the change in n or B due to the non-negligible contribution from the topological edge states in large moiré unit cell area. ν = nA is the number of carriers n per moiré unit cell A. (C) In B-field, ν = 3 state switches between K and K′ valley polarization as doping level changes across the gap. (D) Nonvolatile switching between K and K′ valley-polarized magnetic states independently controlled by either n or B. Panels (A–D) reproduced with permission from ref (828). Copyright 2020 Springer Nature. (E) Temperature-dependent hysteresis (highlighted by colored areas) observed in magnetic field in rhombohedral graphite. (F) Phase diagram of the critical behavior in (E) characteristic to strongly correlated electronic systems. Panels (E, F) are reproduced with permission from ref (832). Copyright 2020 Springer Nature. (G) Scanning tunneling spectroscopy map of a small θ double bilayer graphene showing Bernal (black) and rhombohedral (white) stacking domains. Reproduced with permission under a Creatice Commons CC BY-NC 4.0 license from ref (833). Copyright 2021 National Academy of Sciences. (H–J) Rhombohedral (R) and monoclinic (M) stacking configuration and magnetic domains in a small θ twisted bilayer CrI₃. (H) Three types of stacking domain walls in this system. Arrows represent the stacking vectors. (I) Sketch of the magnon network at θ = 0.1°. (J) Stacking and magnetic domain patterns of the gray rectangle area in (I). Red (blue) arrows represent stacking vectors for R (M) stacking. Red (cyan) lines represent the RR (MM) stacking domain walls. Panels (H–J) are reproduced with permission from ref (834). Copyright 2020 American Physical Society.

Figure 74

Tuning magnetic vdW heterostructures. (A) Schematic of a high-pressure setup for a MTJ. Yellow line represents electrical leads. (B) Tunnel current I_t versus magnetic field H at pressures from 0 to 2.7 GPa in a bilayer CrI₃ MTJ. Insets show spin alignments and optical image of the MTJ. Panels (A, (B) are reproduced with permission from ref (22). Copyright 2019 Springer Nature. (C) Proposed vdW heterostructure where exchange (EX) and spin–orbit (SO) coupling can be swapped by an electric field. Cr₂Ge₂Te₆ magnetization is denoted by red arrows. Adapted with permission from ref (840). Copyright 2020 American Physical Society. (D, E) Dependence of RMCD on magnetic field in heterostructures of WSe₂ and trilayer (D)/bilayer (E) CrI₃ (shown in the insets). Panels (D, E) are reproduced with permission from ref (203). Copyright 2020 Springer Nature. (F–H) Photoemission and spin-dependent charge transfer in hBN encapsulated CrI₃/WSe₂ heterostructure. (F) A schematic of the heterostructure deposited on nanopillar array. (G) Spin-dependent charge transfer from spin-polarized states in WSe₂ to CrI₃ results in the highly p-doped WSe₂, where an exciton can be turned into a localized charged trion (0D X⁺) via a hole capture process (H). Arrows in red and green (blue) denote the spin direction in WSe₂ (CrI₃). Panels (F–H) are reproduced with permission under a Creative Commons CC BY license from ref (841). Copyright 2020 Springer Nature.

Figure 75

Comparison of different Monte Carlo models calculating the Curie temperature of CrI₃. (a) Classical 2D Heisenberg. (b) Classical Ising model. (c) Quantum-like simulation with temperature rescaling. All panels are adapted with permission from ref (865). Copyright 2015 American Physical Society.

Figure 76

Slater-Pauling or volcano plot for 2D magnets. (a) High-throughput screening undertook over several crystal structures and elements of the periodic table including formulas MX′₂, MX, MX′₃, MPX₃, MX₂, and CrFTe₃ with M = Sc–Zn, La, Y; X′=Cl, Br, I; X=O, S, Se, Te; F=Si, Ge. The simulations included mainly transition metals with 3d electrons, but some with 4d and 5d were included for comparison. (b) Variation of the local magnetic moment M(μ_B) at the metal atom as function of its valence Z(e^–). Bader charge analysis was used to extract Z for each metal atom at the compound. The solid lines show a fit to the data set on two different regimes according to the filling of the valence. The positive slope (weak magnets) can be fairly well fitted using M⁺ = 0.84Z – 1.15 (with a linear regression coefficient R² = 0.96) and the negative (strong magnets) with M^– = −0.87Z + 9.27 (R² = 0.90). An electron counting argument can be used to explain both regimes as discussed in the text. (c) Spin resolved density of states (DOS) for monolayer MPTe₃ (M = V, Cr, Mn, Fe, Co, Ni) as function of the energy ε displaying the spin up density n^up (faint gray) and spin down n^down (faint brown) at opposite sides. The energy is shifted to the Fermi energy ε_F at zero. (d) Variation of the model predicted magnetization versus DFT + U calculated magnetization for the compounds showed in (a). Calculations were performed using the VASP code using a 21 × 21 × 1 k-sampling grid, the Dudarev (GGA+U) scheme with Hubbard U values following those in ref (341). The energy cutoff is set to 600 eV, the convergence criteria for energy to 10^–7 eV and for the forces to 0.01 eV/Å. In order to avoid interactions between the layers, we applied periodic boundary conditions with a vacuum space of 25 Å. We used the projector augmented wave (PAW) methods with a plane wave basis. The Vosko–Wilk–Nusair modification scheme is applied for the spin-polarized calculations. All images in this figure are original, and no permissions are required.

Figure 77

Comparison of spin Hamiltonians used to model the magnetic properties of CrI₃. The two most commonly used Hamiltonians in the literature have been the Ising and the Heisenberg (XXZ) models which also includes magnetic anisotropy. The latter was used to understand inelastic tunnelling spectra for MTJs. The former was initially assigned to CrI₃ but its gross overestimation of the Curie temperature relative to experimental data made it unrealistic to account for the interactions in the system. Overall, depending on the property being measured, other alternatives have been considered: (i) for angle-dependent FM resonance measurements, the Kitaev model with quadrupole–quadrupole interactions and the Zeeman coupling was implemented; (ii) for inelastic neutron scattering to extract the magnon dispersion of bulk CrI₃,^, the XXZ model either including DMI or adding biquadratic exchange with DMI have been proposed; and (iii) for the magnetic domains and domain walls on CrI₃, a Hamiltonian taking into account biquadratic exchange was utilized. Starting with image at top, and going clockwise, panels adapted with permission under a Creative Commons CC BY license from refs (796). Copyright 2020 Springer Nature. Adapted with permission from ref (14). Copyright 2018 AAAS. Adapted with permission from ref (745). Copyright 2020 American Physical Society. Reproduce with permission from ref (31). Copyright 2021 John Wiley and Sons. Adapted with permission from ref (746). Copyright 2018 AAAs.

Figure 78

Merons and antimerons on 2D magnet CrCl₃. (a) Artistic view of the presence of topologically nontrivial spin quasiparticles merons and antimerons on monolayer CrCl₃. The different colors follow the orientation of the spins throughout the layer. (b, c) Local views of antimerons and merons, respectively, with their local spin configurations at Cr atoms.